The present invention relates to the quantization of sampled continuous-time analog signals to predefined discrete reconstruction levels for the purpose of representing the analog signals by digital codes and, more particularly, to the quantization of analog signal samples so as to minimize a distortion measure of the form d(u,x)=(Au-Bx).sup.T (Au-Bx).
The quantization of an analog signal according to a predefined set of reconstruction levels can be considered as an optimization problem, in which a sampled continuous-time signal u is approximated by a discrete-amplitude signal x such that a distortion measure d(x,u) is minimized with the constraint that the signal x can only take on values from a predefined set of discrete reconstruction levels. Here u and x are each N-dimensional vectors representing N samples of the continuous-time signal and N corresponding values of the discrete-amplitude signal, respectively, where N is an integer greater than one. Therefore, the values of the quantized signal x approximating the sampled analog signal u depend upon the choice of the distortion measure d(x,u).
In standard analog-to-digital (A/D) conversion, in which each analog signal sample is independently mapped to its closest reconstruction level, the distortion measure d(x,u) being minimized is typically the squared error ##EQU1## where the superscript T, as used in equation (1) and other mathematical expressions in the specification and claims, denotes the transpose of a matrix or vector. This squared error distortion measure has been found to be highly inappropriate for certain applications, such as sound or image coding, where the coded signal is to be ultimately presented for quality evaluation by human sensory perception. For applications involving human sensory perception, a quadratic distortion measure of the form d(u,x)=(Au-Bx).sup.T (Au-Bx), where A and B are appropriately chosen N.times.N matrices, can provide appropriate fidelity criteria for the quantization of signals, since it can represent a frequency-weighted squared error taking into account the frequency dependence of the response of a sensory perception system. Accordingly, a need clearly exists for a nonstandard A/D conversion technique which provides for the quantization of analog signal samples according to a predefined set of discrete reconstruction levels, such that a distortion measure of the form d(u,x)=(Au-Bx).sup.T (Au-Bx) is minimized.
Recently, there has been proposed A/D conversion using a network of nonlinear analog amplifiers which includes a weighting circuit for supplying a weighted sum of the output signals of certain other amplifiers and of certain external signals to the input of each amplifier. The signal supplied to the input of each amplifier is derived by applying a respective weighting factor to each output signal and each external signal to be included in the weighted sum, and taking the sum of the weighted signals.
The foregoing type of network is often referred to as a neural network because such networks are thought to model neural systems of the brains of animals. Neural networks having resistive weighting circuits useful for solving various types of problems are disclosed in U.S. Pat. No. 4,660,166 issued to John J. Hopfield on Apr. 21, 1987. The general structure of a known neural network with a resistive weighting circuit is illustrated in FIG. 1. The circuit 100 has N amplifiers A.sub.1 -A.sub.N each having a normal output x.sub.i and an inverted output x.sub.i, where i=1, 2, . . . , N. The amplifiers each have a nonlinear input-output relation (i.e., transfer function) characterized by a monotonic sigmoid shape, as depicted in FIG. 2. The maximum and minimum levels for the normal output of each amplifier is taken as 0 and 1, respectively, while the corresponding levels for the inverted output is taken as 0 and -1. The sigmoid transfer function of FIG. 2 has a width defined as 2v.sub.0 as shown in the figure.
Referring again to FIG. 1, the outputs of each of the amplifiers A.sub.1 -A.sub.N of the neural network 100 are connected to the inputs of certain other amplifiers through a resistive weighting circuit 101, which comprises an array of resistors, each connecting one of the outputs of a particular amplifier to the input of another amplifier. Each resistor R.sub.ij connecting one of the outputs of an amplifier A.sub.j to the input of another amplifier A.sub.i serves to apply a weighting factor W.sub.ij to an output signal from amplifier A.sub.j before it is summed with other weighted signals and provided to the input of amplifier A.sub.i, where i and j are independent integers having values i=1, 2, . . . , N and j=1, 2, . . . , N. The relative value of each resistor R.sub.ij is defined by the relation R.sub.ij =1/.vertline.W.sub.ij .vertline.. If W.sub.ij is greater than zero, the resistor R.sub.ij is connected to the normal output of amplifier A.sub.j, and if W.sub.ij is less than zero, the resistor R.sub.ij is connected to the inverted output of amplifier A.sub.j. The connections between the inputs and outputs of the amplifiers A.sub.1 -A.sub.N of the neural network 100 may be defined by an N.times.N weighting matrix W having each weighting factor W.sub.ij as an element in a respective row and column thereof.
In addition to a weighted sum of output signals supplied by other amplifiers, each amplifier of the neural network 100 may also receive an external signal I.sub.i forming part of the weighted sum. The external signals I.sub.1 -I.sub.N may be constant signals and/or problem specific input signals.
Connected between the input of each amplifier and a reference ground is an integrating circuit consisting of the parallel combination of a resistor R and a capacitor C. The values of R and C are chosen such that the time constant of each integrating circuit is appropriate for taking the analog sum of the weighted output signals and external signals applied to the input lines l.sub.1 -l.sub.N extending from the inputs of amplifiers A.sub.1 -A.sub.N, respectively.
Where the diagonal elements, W.sub.ii, of the weighting matrix W are all zero and the amplifier A.sub.1 -A.sub.N each have a high gain (i.e., a sharp sigmoid transfer function), the stable states of the neural network 100 are the local minima of the quantity. ##EQU2##
The use of neural networks of the type illustrated in FIG. 1 for standard A/D conversion is described in a paper entitled "Sample `Neural` Optimization Networks: an A/D Converter, Signal Decision Circuit, and a Linear Programing Circuit" by David W. Tank and John J. Hopfield, publisehd in the IEEE Transactions on Circuits and Systems, VOL. CAS-33, No. 5, May 1986, pp. 533-541. For the standard A/D conversion problem, equation (2) takes the form of ##EQU3## where u.sub.n is a sample of an analog signal being digitized. According to equation (3), each weighting factor W.sub.ij has the value of -2.sup.(i+j) and each external signal T.sub.i has the value -2.sup.(2i-1) +2.sup.i u.sub.n.
An example of a 4-bit standard A/D converter implemented with a neural network having a resistive weighting circuit in accordance with the aforementioned D. W. Tank et. al paper is shown in FIG. 3. A continuous-time analog signal u(t) is provided to a sample-and-hold circuit 301, which samples u(t) at a rate determined by a sampling clock signal .alpha..sub.s to provide analog signal samples u.sub.n. The analog signal samples u.sub.n are provided one at a time to the resistive weighting circuit 302 of a neural network 300 having four amplifiers A.sub.0 -A.sub.3. A negative one-volt constant signal is also applied to the resistive weighting circuit 302. The relative values of the resistance of the weighting circuit 302 are determined according to the foregoing relationships for the weighting factors W.sub.ij and external signals I.sub.i.
The bit values b.sub.0, . . . b.sub.3, of the digital code representing each analog signal sample u.sub.n are provided at the normal output terminals of the amplifiers A.sub.0 -A.sub.3, respectively.
Analog-to-digital converters of the type illustrated in FIG. 3 have the drawback in that the circuit 300 does not always stabilize to an acceptably accurate digital representation of the analog signal sample u.sub.n. Furthermore, since such circuits provide for standard A/D conversion, in which each analog signal sample is mapped to its closest reconstruction level, the quantized digital signals provided thereby are not optimal for use in certain applications, such as those involving human sensory perception.